A104496 Expansion of 2*(2*x+1)/((x+1)*(sqrt(4*x+1)+1)).

1, 0, 0, -1, 5, -19, 67, -232, 804, -2806, 9878, -35072, 125512, -452388, 1641028, -5986993, 21954973, -80884423, 299233543, -1111219333, 4140813373, -15478839553, 58028869153, -218123355523, 821908275547, -3104046382351, 11747506651599, -44546351423299, 169227201341651

Offset: 0

Author

Paul D. Hanna, Mar 11 2005

Keywords

• sign
• easy

Comments

Previous name was: Row sums of triangle A104495. A104495 equals the matrix inverse of triangle A099602, where row n of A099602 equals the inverse Binomial transform of column n of the triangle of trinomial coefficients (A027907).

Absolute row sums of triangle A104495 forms A014137 (partial sums of Catalan numbers).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A104495, A099602, A027907, A000108.

Programs

Maple

gf := (2*(2*x+1))/((x+1)*(sqrt(4*x+1)+1)): ser := series(gf,x,30):
seq(coeff(ser,x,n),n=0..28); # Peter Luschny, Apr 25 2016

Mathematica

CoefficientList[Series[(1+2*x)/(1+x)/(1+x - (1-(1+4*x)^(1/2))^2/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2014 *)

PARI

{a(n)=local(X=x+x*O(x^n));polcoeff( (1+2*X)/(1+X)/(1+X-(1-(1+4*X)^(1/2))^2/4),n,x)}

Python

from itertools import accumulate
def A104496_list(size):
    if size < 1: return []
    L, accu = [1], [1]
    for n in range(size-1):
        accu = list(accumulate(accu + [-accu[0]]))
        L.append(-(-1)**n*accu[-1])
    return L
print(A104496_list(29)) # Peter Luschny, Apr 25 2016

Formula

G.f.: A(x) = (1 + 2*x)/(1+x)/(1+x - x^2*Catalan(-x)^2), where Catalan(x)=(1-(1-4*x)^(1/2))/(2*x) (cf. A000108).

a(n) ~ (-1)^n * 2^(2*n+1) / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2014

D-finite with recurrence: (n+1)*a(n) +(7*n-3)*a(n-1) +2*(7*n-12)*a(n-2) +4*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jan 23 2020

Extensions

New name using the g.f. of the author by Peter Luschny, Apr 25 2016